Definition 2.5 (see [[1]}, Section 3] and [[2]}, Definition 2.16])
Let \(\mathcal {X}\)  be a projective Deligne-Mumford stack over \(k\)  with a moduli scheme \(X\) . Suppose \(\mathcal {E}\)  is a generating sheaf for \(\mathcal {X}\)  and \(\mathcal {O}_{X}(1)\)  is a very ample invertible sheaf on \(X\)  relative to \(\mathrm {Spec}\,k\) , we define \((\mathcal {E}, \mathcal {O}_{X}(1))\)  as a polarization of \(\mathcal {X}\) .
